where \(g\)  is the gravity constant, \(H\)  is the mean active layer depth and \(\omega _{c}\)  is the Coriolis frequency, assumed to be constant. Notice that the case \(\lambda =0\)  corresponds to the velocity-vorticity formulation of Euler equations.
The system (REF ) is commonly used to track the dynamics of the atmospheric and oceanic circulation at large scale motion. For a general review about the asymptotic derivation of the these equations from the rotating shallow water equations we refer for instance to [1]}.
